Skip to content

Structural Stability and Dynamics

This document collects implemented stability diagnostics, phase-transition read-outs, life telemetry, lifecycle classification, an auxiliary Hamiltonian model, and integrity tools anchored to the nodal equation \(\partial\mathrm{EPI}/\partial t = \nu_f \cdot \Delta\mathrm{NFR}(t)\). Exact restricted results, selected policies, finite observations, and compatibility models are identified separately.

Status: Mixed — exact restricted diffusion results, operational operator classifications and measured diagnostics are identified separately.

Ownership. This is a capability and scope reference, not a research queue. The diffusion theorem and REMESH history theorem own their respective derivations; operator-event contracts own execution and evidence admission. Sections 1.4-1.6 retain the applicability bridges and a held/refreshed-pressure counterexample. The field-energy balance belongs to the conservation note, and phase/telemetry interpretations to parameter foundations. Current research priorities belong to the single execution plan.

1. Lyapunov Stability Analysis

1.1 Tetrad-plus-current structural energy

compute_energy_functional implements the non-negative quadratic diagnostic

\[ E[G] = \frac{1}{2}\sum_i \left[\Phi_s(i)^2 + |\nabla\phi|(i)^2 + K_\phi(i)^2 + J_\phi(i)^2 + J_{\Delta\mathrm{NFR}}(i)^2\right] \]

This is the tetrad-plus-current structural energy: it contains three tetrad fields \((\Phi_s,|\nabla\phi|,K_\phi)\) and the two currents \((J_\phi,J_{\Delta\mathrm{NFR}})\). It omits the fourth tetrad field \(\xi_C\), so calling it the “tetrad energy” is inaccurate. It also contains no explicit EPI or \(\nu_f\) term, although recomputing the fields after a state change can alter the value indirectly.

For general engine trajectories, \(E\) is a Lyapunov candidate. The finite difference returned by compute_lyapunov_derivative only reports whether two supplied snapshots show non-increase within its tolerance. Grammar compliance does not prove \(dE/dt\leq0\). A proved Lyapunov result exists for the different, weighted centered energy of restricted pure EPI diffusion; see the heterogeneous diffusion theorem.

The historical coherence_matrix is also outside this Lyapunov claim. It is an auxiliary entrywise-bounded affinity built from phase, EPI, frequency and Si, and it is distinct from canonical \(C(t)\). Entrywise nonnegativity does not imply a nonnegative spectrum after the graph-support mask: three identical nodes on a path give \(W=I+A_{P_3}\) with eigenvalue \(1-\sqrt{2}<0\). The matrix can supply a Hermitian term to the auxiliary Hamiltonian because it is real symmetric; it is not a positive-semidefinite coherence observable.

1.2 Canonical channels and nominal U2 roles

The generated canonical contract table owns primary nodal channels, tokens and measurement contexts; the separate generated grammar role table owns U2 composition roles. A primary channel is not a list of every secondary write or a stability classification: phase-channel ZHIR is a U2 destabilizer, while pressure-channel NAV is U2-neutral.

physics.lyapunov maps those U2 roles to a legacy compatibility model. Its public objects retain names such as OperatorLyapunovBound, but the multipliers are nominal policy values, not proved bounds on the structural energy above or on \(C(t)\). The preferred API names are OperatorPolicyMultiplier, OPERATOR_POLICY_MULTIPLIERS, U2PolicyRole, evaluate_sequence_policy, and compare_operator_energy_to_policy; the legacy names remain source-compatible wrappers.

Stabilizer policy entries

Operator Canonical default input Policy multiplier
IL (Coherence) pressure retention \(\pi/(\pi+1)\approx0.7585\) \(0.7585\)
THOL (Self-organization) acceleration \(1/(4\pi)\approx0.0796\) \(1-1/(4\pi)\approx0.9204\)

Destabilizer policy entries

Operator Canonical default input Policy multiplier
OZ (Dissonance) pressure scale \((\pi+1)/\pi\approx1.3183\) \(1.3183\)
ZHIR (Mutation) phase-shift factor \(1/\pi\approx0.3183\) \(1+1/\pi\approx1.3183\)
VAL (Expansion) capacity scale \(1+1/(4\pi)\approx1.0796\) \(1.0796\)

U2-neutral policy entries

AL, EN, UM, RA, SHA, NUL, NAV, and REMESH receive multiplier one because they carry no U2 stabilizer/destabilizer debt. This value does not predict zero observed change in phase-dependent or pressure-dependent energy fields.

The legacy compute_operator_energy_bound and compute_sequence_energy_bound functions interpret their numeric input as an abstract policy score. Their names do not confer energy-bound semantics. The legacy within_bound result of verify_operator_lyapunov is a one-sided screen against that model and carries is_lyapunov_certificate=False. Actual field-energy and coherence changes require before/after telemetry, including the actual glyph executed after grammar selection. See example 39.

1.3 U2 stability scope

Grammar rule U2 requires destabilizers to be accompanied by stabilizers and limits uncompensated debt. This is an operator-composition policy. It does not by itself imply

\[ \sum_{\text{ops}} \Delta E_{\text{op}} \le 0 \]

because the grammar does not supply a common state functional, elapsed times, or proved gain bounds for arbitrary operator realizations. Even a nominal multiplier product at most one remains a compatibility-model result; analytic contractivity requires a separate dynamical proof.

Examples 29 and 38 report the nominal product beside measured energy changes. Agreement or disagreement in a finite run measures the compatibility model; it cannot establish a grammar-wide guarantee.

analyze_operator_policy_context also reports a U2 multiplier beside the normalized graph diffusion gap. It deliberately leaves the historical effective_convergence_rate field as NaN: an operator-position multiplier and a continuous-time pure-EPI eigenvalue cannot be combined without an explicit operator-time model. Its policy_half_steps is a score-model statistic; its diffusion_relaxation_time is the separate unit-capacity pure-EPI scale.

1.4 Restricted affine EPI gain theorem

The affine-reset and hybrid-word theorem supplies a conditional gain independently of U2 policy. For one positive common diffusion metric H=diag(h), let Q=I-1h^T/(h^T1) and V(x)=||Qx||_H^2/2. A declared reset x+=Ax+b has a finite global multiplicative V gain exactly when A1 and b are uniform. Otherwise a uniform input gives the decisive V_before=0<V_after counterexample. For an admitted map, the sharp gain is the squared weighted induced norm of QAQ.

The owner derives the represented-coefficient rational upper bound and its composition with certified continuous diffusion. Neither an eigensolver estimate, a caller tolerance nor a nominal policy multiplier decides contraction. A finite-horizon bound needs valid reset/flow proofs in one exactly matched normalized metric and node order. Declared repetition with positive flow duration lets the contraction bound certify asymptotic disagreement decay. It identifies the limit with the initial weighted consensus when the reset and flow proofs also establish exact preservation of that mean.

The EN/RA realization boundary keeps four objects separate: an ideal-real convex blend, its represented coefficient matrix, the two-stage binary64 proposal, and the accepted runtime snapshot. Its fixed connected undirected support, positive capacities, signed scalar or uniform-real EPI, convex mixing and inactive hard-clipping premises must all hold. RA additionally requires admitted circular U3 neighbours and preserved sign/kind identity. A nontrivial local blend need not preserve the weighted mean, and a capacity boost can change the diffusion metric. Fixed-map repetition requires fixed support and, for RA, fixed U3 sets and a forward-invariant sign/kind domain. Snapshot agreement does not prove global binary64 affinity.

Atomicity is a separate operator execution contract. A stage derives all proposals from one immutable snapshot and validates them before committing graph-owned state; ordered auxiliary streams and opaque callbacks retain their stated limits. This supplies no affine gain by itself. The event-stage composition contract owns optional represented-map certificates and explicit abstentions. Only intact maps bound to the observed EPI endpoints and one common metric enter its finite product. In particular, the REMESH glyph is advisory; delayed EPI mixing is a separate history-dependent operation.

1.5 Operator-event physical-time boundary

The schedule and transaction contract assigns zero duration to each named jump and m+1 declared flow intervals to a word of m events. Exact rationalized binary64 durations and offsets determine ordering; absolute float timestamps are displays. This clock distinguishes flow duration from operator counts and supplies no laboratory calibration. The executor's finite held-input identity, graph rollback and external-effect limits are specified there; none establishes solver accuracy or order.

Held and refreshed pressure define different models. For fixed pure-EPI A=diag(nu_f)L_rw and T=sum_k h_k, the exact-real outer maps are

\[ \widehat{x}_H=(I-TA)x_0, \qquad \widehat{x}_R=\left[\prod_k(I-h_kA)\right]x_0. \]

Internal substeps that retain the initial pressure leave the first map unchanged; explicit pressure refresh gives the second. On the nonuniform mode of unit-capacity \(K_2\), \(\mu=2\), \(T=1.25\) gives \(g_H=-1.5\), whereas two refreshed segments of length \(0.625\) give \(g_R=0.0625\). Thus even modal stability can change. This counterexample concerns the real-arithmetic maps, not a claim that an observed binary64 endpoint equals either formula.

Captured flow evidence distinguishes a measured nodal residual, trusted held-pressure replay and exact affine identification. Pressure-refreshed partitions add the actual refresh boundaries. Exact endpoint promotion requires fixed conductance/capacity, verified pure-EPI pressure and intact per-segment evidence; modal agreement alone is insufficient. Equal spectra do not establish a common eigenbasis or commuting generators. Missing or failed evidence remains an abstention rather than a proof of the exact map.

The Mutation refinement contract separates an offline coordinate pairing from a jump bound to the same execution. Only the terminal live segment supplies Mutation's observed secant; a whole-parent secant is a different observation. The sufficient held-pressure gate certificate requires strict rate-error margin and observed decision agreement. Physical refresh can change the gate legitimately. These observations do not prove U4 readiness, adaptive grammar, solver order or future behavior; §6.2 separates trigger evidence from the nodal-product prediction.

Delayed REMESH requires augmented history. A fixed-history zero coefficient is not a repeated gain: with lag one and alpha=1, EPI (2,0) and delayed row (0,2) alternate under successive empty-schedule cycles. The recorded/causal sequence contract retains this obstruction and distinguishes caller-ordered endpoint continuity from same-invocation causal provenance and graph-owned atomicity.

For fixed uniform alpha in [0,1], positive delays, ordered support and a positive diagonal spatial metric, the separate unclipped exact companion-history theorem uses the stationary temporal weights in V(X)=sum_j pi_j ||Q_H x_j||_H^2/2. Its Jensen balance proves nonincrease; interior alpha gives temporal convergence to each coordinate's history barycenter, while alpha=1 admits periodic histories. This is neither spatial consensus nor zero pressure. The runtime history balances retain signed rounding/clipping defects and the difference between a lifted companion state and the live history. Hard-clamp nonexpansiveness does not control preceding rounding; soft clipping can increase disagreement. Finite adjacent-cycle balances telescope only after their history, metric and endpoint bindings pass.

A positive normalized margin on one causal block does not establish a uniform repeated-runtime theorem. Normalization is undefined at zero initial energy, and amplitude scaling excludes a uniform positive absolute drop. Promotion through these block bounds requires a forward-invariant runtime class, a uniform positive normalized margin and an intrablock prefix bound. The exact common-gain and signed-defect theorems supply conditional bounds for declared schedule families: fixed companion and metric, consensus preservation, common gain q in [0,1], and, for the robust extension, E_H(z)-E_H(y)<=eta*J with eta>=0 and Jensen input envelope J. This signed defect differs from the energy of z-y. q_eff=q*(1+eta)<1 gives strict decay; equality only gives nonincrease. Finite runtime verification of these inequalities does not prove class invariance or future repetition.

The binary64 REMESH class certificates retain the large-rounding counterexample, exact alpha=1 class and sharp antisymmetric P2 half-alpha bound. Their restricted numeric domains are not a complete graph/event model. P2 Reception/REMESH binding separately links the half-Reception q=0 kernel to accepted finite stages, causal sequences and individually transactional policy invocations. Each call is revalidated; finite extinction of active-history disagreement does not certify auxiliary Reception state, future calls or full TNFR stability.

Finally, three-mesh and reversible-mode references separate finite comparisons from the exact reversible single-eigenmode Euler theorem. Its fixed reversible generator, positive capacity, nonzero single mode and 0<mu*h<1 proper-subdivision premises are essential. The executor binding propagates represented defects in the full state, since they can leave that mode. The effective-P2 REMESH family adds its own zero-defect, delay and hard-clipping hypotheses. Neither a finite error decrease nor endpoint binding proves binary64 asymptotic convergence, solver order, repeated mixed dynamics or future stability.

The executable 165_operator_event_relaxation.py records the schedule-only distinctions with two coincident jumps. Continuous duration and Euler modal counts do not redefine U2 debt or U4 recency.

1.6 Spectral Gap Characterisation

For connected fixed symmetric pure EPI diffusion, the first positive generalized eigenvalue \(\lambda_*\) of \(Bv=\lambda Hv\), with \(H=\operatorname{diag}(d_i/\nu_i)\), controls the weighted-energy decay:

Quantity Expression Physical meaning
Energy decay \(V(t) \le e^{-2\lambda_*t}V(0)\) Exact upper bound
Relaxation scale \(1/\lambda_*\) Heterogeneous structural time
Homogeneous reduction \(\lambda_*=\nu_f\lambda_2(L_{sym})\) Common-capacity case

The corresponding exact single-mode Euler theorem is now available for every fixed connected symmetric rational conductance with positive rational capacity. If v=x_0-mean_H(x_0)*1 is nonzero and satisfies the exact identity A*v=mu*v for A=diag(nu_f)L_rw, then

x(T) = mean_H(x_0)*1 + exp(-mu*T)*v,
x_P(T) = mean_H(x_0)*1 + product_j(1-mu*h_j)*v.

For a positive partition with 0 < mu*h_j < 1, the factor error is bounded by (mu^2/2) sum_j h_j^2 <= (mu^2/2) T h_max. Multiplication by ||v||_inf gives the exact L_inf endpoint error; multiplication by E_H(v) times the squared factor error gives its H-error energy. Proper positive subdivision strictly improves the factor and quadratic bound, and the h_max estimate proves conditional exact-real convergence for fixed-data admissible families. The complete derivation and runtime-scope boundary are centralized in the reversible single-eigenmode theorem. The pure certificate does not inspect binary64 execution, glyphs or REMESH.

This table is the exact real-arithmetic theorem. The executable binary64 certificate separately rationalizes its materialized generator and displayed metric. It first requires exact invariance of the consensus subspace, then proves positivity of the symmetrized dissipation on the metric-orthogonal quotient and returns a downward-rounded certified rate. Its generalized eigenvalue remains an estimate. Exact conservation of the displayed weighted mean is a further independent Boolean; without it, the snapshot projection center is not promoted as the final consensus value.

For time-varying capacities satisfying positive finite per-node bounds, the Dirichlet energy is a common Lyapunov function with decay rate at least 2 lambda_2(B) min_i(lower_i/d_i). The executable theorem treats the effective binary64 conductances and capacity bounds as exact real coefficients, then forms degrees, the Laplacian, the quotient gap and the rate in rational arithmetic. Ordinary binary64 eigengaps, products and energies remain labelled estimates; whether diag(strength)-adjacency happens to annihilate constants after floating accumulation is a separate diagnostic. A positive exact rate that underflows on publication still proves the exact-real theorem but supplies no operational binary64 rate. The routine neither observes the future capacity schedule nor certifies a numerical integrator. Within the conditional exact-real model the field converges to consensus, but the consensus value is schedule-dependent unless the capacity ratios remain fixed.

For a finite family of changing symmetric topologies on fixed node support, the normalized metric d_i/nu_i is a common Lyapunov metric when the raw represented metric vectors are exactly proportional across every regime. Caller-tolerance proximity is diagnostic and cannot promote the theorem. The minimum certified rational quotient bound then controls arbitrary switching. Every represented regime must fix the uniform EPI field exactly; the weaker consensus-subspace condition and preservation of the displayed weighted mean are reported independently and cannot replace that canonical fixed-point identity. The object's equilibrium, Lyapunov-value and derivative fields are binary64 diagnostics for its first snapshot and displayed normalized metric. Composition instead uses the rationalized reference metric and certified rate; a sampled trajectory must recenter that common metric at every state. This is a restricted topology-change theorem; it does not cover node creation/removal or nodal-type transitions.

Implementation: physics.structural_diffusion.verify_heterogeneous_diffusion_stability provides the exact fixed-capacity certificate; derive_time_varying_diffusion_stability_bound supplies the conditional common bound; verify_switching_diffusion_stability checks the common-metric switching theorem. diagnose_euler_relaxation_window resolves the frozen graph's actual explicit-Euler modal factors and solver-step relaxation count, using a dimensionless zero-mode tolerance relative to the fastest decay rate. That count is a numerical-integration quantity, not the U4 operator-position window. physics.lyapunov retains nominal operator-role diagnostics and must not be read as a universal convergence prover.

2. Phase Transitions

2.1 Order-parameter candidate

The symmetry-breaking field \(\mathcal{S}\) is the implemented order-parameter candidate for controlled phase-transition sweeps:

\[ \mathcal{S}(i) = \left(|\nabla\phi|^2 - K_\phi^2\right) + \left(J_\phi^2 - J_{\Delta\mathrm{NFR}}^2\right) \]

2.2 Phase Classification

The phase is decided by the standardized spatial imbalance of the signed global mean, \(z = |\langle\mathcal{S}\rangle| / \sqrt{\mathrm{Var}(\mathcal{S})/N}\), and likewise for signed chirality. Because graph nodes are coupled, this ratio is not a hypothesis-test z-score without an independent sampling or effective-sample-size model. The engine retains the historical function name symmetry_zscore, but uses the ratio only as a deterministic classifier input. The selected operational cut is \(z = 1\). The local magnitudes \(\langle|\mathcal{S}|\rangle\) and \(\langle|\chi|\rangle\) remain separate telemetry and cannot establish global symmetry breaking because opposite signs may cancel.

Phase Condition Operational meaning
NON_LIFE \(z \le 1\) signed mean below the selected standardized-imbalance cut
LIFE \(z > 1\) AND \(z_\chi > 1\) both signed-imbalance ratios exceed the selected cut
CRITICAL \(z > 1\) AND \(z_\chi \le 1\) order imbalance without a chirality-ratio crossing

The classifier evaluates the ratio in scaled coordinates before dimensional variance can underflow. Snapshot and series share that capture path and retain coherence-length availability/provenance; unavailable xi is not rewritten to zero. Unrepresentable variance or susceptibility rejects explicitly.

A negative control demonstrates the size boundary: on P3 with phases [0, 0.2, 0.6] and stored pressure [0, 0.3, 0.6], the label is NON_LIFE. The disjoint union of two identical copies is labelled LIFE: the local fields and component dynamics are unchanged, but both ratios acquire a factor sqrt(2). This is the declared normalization, not autonomous formation. No production selector consumes these labels. The phase controls retain this witness.

2.3 Effective time-series exponent fit

fit_critical_exponent and detect_phase_transition fit the diagnostic model

\[ |\langle\mathcal{S}\rangle| \sim |t-t_c|^{p_{\mathrm{fit}}} \]

by log-log regression. In the time-series detector, \(t_c\) is the time of the largest sampled susceptibility, when that maximum is positive. The fit uses only positive samples strictly after \(t_c\); it never substitutes earlier samples. It returns exponent/measured_exponent and \(R^2\), or None when fewer than three eligible post-\(t_c\) samples exist.

Positivity, rather than a fixed amplitude/time epsilon, determines eligibility. Centered shared correlation supplies the fit quality; a tiny imperfect response does not become a perfect fit through an absolute residual cutoff. An initially above-threshold sample takes precedence over a later recrossing when reporting the first observed crossing.

The neutral symbol \(p_{\mathrm{fit}}\) avoids assigning competing \(\beta\) and \(\gamma\) names to the same implemented regression. The result is protocol-dependent. It becomes a conventional critical exponent only when the time coordinate is mapped to a declared control-parameter distance and a finite-size protocol supports that interpretation. No universal value follows from the nodal equation.

2.4 Classification scale

The classifier uses only Z_SIGNIFICANCE = 1, a selected standardized-spread policy. It is neither a p-value cut nor a graph-independent critical constant. Legacy constants such as \(\pi/16\), 0.034 and 0.155 do not enter this phase classification and must not be presented as universal transition thresholds.

2.5 Susceptibility

The finite-sample structural susceptibility diagnostic is

\[ \chi_{\mathcal{S}}(t) = N \cdot \operatorname{Var}(\mathcal{S}) \]

The implementation records its maximum along the supplied sequence. A sampled peak is not a divergence theorem.

2.6 Finite-size protocol

analyze_phase_finite_size_scaling accepts one shared control grid at three or more node counts, with a balanced replicate axis. It reports the sampled pseudocritical control, replicate standard errors and power-law slopes of peak susceptibility, order magnitude and coherence length against node count N. The use of N is explicit: converting these slopes to conventional exponent ratios requires an independently justified linear-size or dimension map. Replicating one graph family does not establish universality. A susceptibility maximum on the first or last sampled control value is flagged as unbracketed instead of being presented as a located critical point. Every exact maximizer is retained, so plateaus are marked ambiguous and any boundary contact makes the sampled peak unbracketed.

Implementation: src/tnfr/physics/phase_transition.py provides snapshot and time-series classification; src/tnfr/physics/phase_scaling.py provides the balanced finite-size diagnostic.

3. Life-telemetry diagnostics

3.1 Implemented formulas

physics.life consumes supplied time series; it does not evolve the graph. It is an assumption-explicit logistic diagnostic, not a biological classifier or an operator implementation. Every channel must be a finite numeric one-dimensional series; Boolean, multidimensional, NaN and infinite inputs are rejected. detect_life_emergence additionally requires a nonempty nonnegative EPI series, matching channel shapes and strictly increasing sample times. It requires \(0\leq\varepsilon\leq1\), \(\gamma\geq0\), and \(\mathrm{EPI}_{\max}>0\). No input is clipped or broadcast implicitly.

Let \(x(t)\geq0\) be the supplied EPI-magnitude series. The declared model computes

\[ G(t)=\gamma x(t)\left(1-\frac{x(t)}{\mathrm{EPI}_{\max}}\right) \]

and the time-local autopoietic coefficient

\[ A(t)=\frac{G(t)\,\dot x(t)}{|\Delta\mathrm{NFR}_{\mathrm{ext}}(t)|^2+\epsilon_{\mathrm{num}}}. \]

The numerator and denominator are evaluated element by element. The implementation does not take the ensemble or time averages shown in older versions of this note, and dEPI_dt supplies \(\dot x\) rather than the function estimating it internally. The small \(\epsilon_{\mathrm{num}}\) is the shared safe division guard.

The remaining returned series are

\[ V_i(t)=\frac{|\varepsilon G(t)|} {|\varepsilon G(t)|+|\Delta\mathrm{NFR}_{\mathrm{ext}}(t)|+\epsilon_{\mathrm{num}}}, \]
\[ S(t)=\frac{\varepsilon\,|\gamma(1-2x(t)/\mathrm{EPI}_{\max})|} {|\partial_t\Delta\mathrm{NFR}_{\mathrm{ext}}(t)|+\delta+\epsilon_{\mathrm{num}}}, \qquad M(t)=\frac{x(t)-\mathrm{EPI}_{\max}/2}{\mathrm{EPI}_{\max}}. \]

LifeTelemetry.vitality_index is the internal-versus-total pressure ratio \(V_i\) above. It does not include \(C(t)\); callers may combine the two as a separate analysis.

3.2 Operational threshold detection

detect_life_emergence uses \(A(t)>1\) as a selected operational event. It linearly interpolates the first transition from \(A\leq1\) to \(A>1\), returns the first supplied time when the series already starts above one, and otherwise returns None. This classifier records a telemetry crossing; it does not prove future self-sustenance or a biological classification.

The returned LifeTelemetry contains the supplied times, \(V_i\), \(A\), \(S\), \(M\), and life_threshold_time.

4. Node lifecycle classifier

4.1 Implemented state priority

get_lifecycle_state is an instantaneous rule-based classifier. It does not estimate whether \(\nu_f\) or \(\Delta\mathrm{NFR}\) is increasing. With the default parameters, it evaluates conditions in this order:

Returned state Implemented condition
COLLAPSING \(\nu_f<0.01\), or \(\lvert\Delta\mathrm{NFR}\rvert>10\), or a non-isolated node has coupling \(<0.1\)
MUTATION \(\lvert\Delta\mathrm{NFR}\rvert>5\) and \(\nu_f>0.1\)
PROPAGATION coupling \(>0.7\) and \(\nu_f>0.1\)
STABILIZATION \(\lvert\Delta\mathrm{NFR}\rvert<1\) and local structural coherence \(>0.8\)
ACTIVATION \(\nu_f\geq0.1\) after the earlier checks
DORMANT all remaining states above the collapse-frequency cut

The stabilization test reads the shared local diagnostic structural_coherence(DeltaNFR, dEPI_dt) = 1/(1 + |DeltaNFR| + |dEPI_dt|). It uses the stored pressure and rate, not EPI magnitude or a prediction reconstructed from the nodal product. The threshold is a configured classifier policy, not a persistence theorem. LifecycleState.COLLAPSED exists in the enum but get_lifecycle_state currently returns COLLAPSING for every collapse trigger and never returns COLLAPSED.

For a node with neighbors, the coupling diagnostic uses the shared circular neighbor mean and shortest-arc phase difference:

\[ \bar\theta_i=\operatorname{atan2}\!\left(\sum_{j\in\mathcal N(i)}\sin\theta_j, \sum_{j\in\mathcal N(i)}\cos\theta_j\right),\qquad c_i=1-\frac{|\operatorname{wrap}(\theta_i-\bar\theta_i)|}{\pi}. \]

The numerical mean convention does not establish a uniquely defined direction when the neighbor resultant vanishes. Isolates receive \(c_i=0\), and the network-decoupling collapse check applies only when neighbors exist.

4.2 Collapse-reason check

check_collapse_conditions is a separate predicate. It returns the first matching reason in this order:

Collapse reason Default condition
Frequency failure \(\nu_f<0.01\)
Extreme dissonance \(\lvert\Delta\mathrm{NFR}\rvert>10\)
Network decoupling non-isolated node with \(c_i<0.1\)
EPI dissolution scalar EPI \(<0.01\)

The EPI-dissolution condition belongs to this separate predicate and is not consulted by get_lifecycle_state. Configuration values override graph values, which in turn override these operational defaults.

Implementation: operators.lifecycle provides LifecycleState, CollapseReason, get_lifecycle_state, check_collapse_conditions, and should_collapse.

5. Auxiliary internal Hamiltonian

5.1 Implemented matrix

operators.hamiltonian constructs the finite matrix

\[ H_{\mathrm{int}}=H_{\mathrm{coh}}+H_{\mathrm{freq}}+H_{\mathrm{coupling}}, \]

with the literal implementation

\[ H_{\mathrm{coh}}=C_0 W,\qquad H_{\mathrm{freq}}=\operatorname{diag}(\nu_{f,1},\ldots,\nu_{f,N}),\qquad H_{\mathrm{coupling}}=J_0 A_{\mathrm{sym}}. \]

\(W\) is the matrix returned by coherence_matrix. The constructor default is \(C_0=-1\), so the coherence term is \(-W\); writing an additional leading minus sign reverses the implemented sign. The default coupling is \(J_0=0.1\), and the builder writes both matrix directions for every graph edge. The constructor requires finite entries in every component and their sum, then checks an absolute Hermiticity residual tolerance of 1e-10. The auxiliary scale must be a finite nonzero real number. Acceptance is not an exact algebraic Hermiticity certificate.

This matrix supplies an auxiliary linear model. The repository does not derive the general engine trajectory or the canonical graph \(\Delta\mathrm{NFR}\) from it.

5.2 Unitary flow and eigenmodes

For an exactly Hermitian matrix and real nonzero declared \(\hbar_{\mathrm{str}}\), the ideal evolution is unitary:

\[ U(t)=\exp\left(-\frac{iH_{\mathrm{int}}t}{\hbar_{\mathrm{str}}}\right). \]

time_evolution_operator numerically evaluates this exponential and applies an allclose unitarity check. get_spectrum uses a Hermitian eigensolver; its returned ascending eigenvalues and eigenvectors numerically approximate

\[ H_{\mathrm{int}}|\phi_n\rangle=E_n|\phi_n\rangle. \]

For the exact Hermitian model, an eigenvector evolves only by the phase \(e^{-iE_nt/\hbar_{\mathrm{str}}}\) in this auxiliary unitary flow. That makes it a stationary ray of this model; it does not make it a maximally stable TNFR configuration or establish dissipative attraction.

5.3 Compatibility helpers and sign scope

Despite its name, compute_delta_nfr_operator() does not compute a commutator. It literally returns

\[ G_+=\frac{i}{\hbar_{\mathrm{str}}}H_{\mathrm{int}}, \]

which is anti-Hermitian and is the negative of the ket-state generator \(-iH_{\mathrm{int}}/\hbar_{\mathrm{str}}\) used by \(U(t)\).

compute_node_delta_nfr(n) implements the localized-projector read-out for \(\rho_n=|n\rangle\langle n|\), defined by the real part of

\[ \frac{i}{\hbar_{\mathrm{str}}} \langle n|[H_{\mathrm{int}},\rho_n]|n\rangle. \]

For this localized projector, the displayed diagonal commutator is exactly zero: \([H,\rho_n]_{nn}=H_{nn}-H_{nn}=0\). The implementation returns this identity directly after checking captured node membership, without matrix products. Under the implemented unitary \(U\rho U^\dagger\), the density-matrix derivative would instead carry the sign \(-i[H,\rho]/\hbar_{\mathrm{str}}\). These helpers therefore do not reconstruct the engine's node-local structural pressure; canonical \(\Delta\mathrm{NFR}\) is computed by dynamics.dnfr.

Selecting its opt-in compute_delta_nfr_hamiltonian hook writes zero stored pressure at every node and labels that null read-out in pressure metadata. It supplies no density evolution or independently derived pressure law. The hook rebuilds its auxiliary matrix on each invocation, so capacity, support, affinity configuration and scale changes are not hidden by a node-count cache. cache_hamiltonian=True retains only the latest constructed snapshot for inspection; it does not reuse that snapshot on the next call. Regression coverage belongs to test_hamiltonian_pressure_scope.py.

6. Optional structural integrity tools

6.1 Reactive monitor

The reactive monitor is opt-in. enable_integrity_monitor(G, mode=...) creates a StructuralIntegrityMonitor and stores it in G.graph["integrity_monitor"]. Calls through the operator-class pipeline then invoke before_operator and after_operator. Without an attached monitor, ordinary operator calls do not run these diagnostics.

For each monitored call, the implementation attempts to compare conservation snapshots, the finite change of the structural-energy candidate, heuristic grammar-violation labels, Noether-charge drift, and an operator-specific postcondition. These are runtime diagnostics. They do not turn the five-term energy into a general Lyapunov theorem. ENFORCE raises after a reported unhealthy result; it does not roll back an operator mutation.

An IntegrityReport is healthy only when its conservation quality is above 0.7, its sampled energy change is classified stable, no grammar diagnostics are present, and its postcondition check passes. Charge drift is reported but is not part of that property.

6.2 Implemented postcondition registry

POSTCONDITIONS has one entry for every canonical operator name, but several entries check only a measurable proxy and REMESH is explicitly advisory:

Operator Check currently performed by the reactive registry
AL EPI does not decrease; \(\nu_f\), phase and \(\Delta\mathrm{NFR}\) do not change
EN immediate operator-local \(C(t)\) is unchanged before pressure refresh
IL \(C(t)\) does not decrease and \(\lvert\Delta\mathrm{NFR}\rvert\) does not increase
OZ \(\lvert\Delta\mathrm{NFR}\rvert\) does not decrease
UM \(\lvert\Delta\mathrm{NFR}\rvert\) does not increase
RA nonzero EPI sign is preserved and \(\nu_f\) does not decrease
SHA EPI is unchanged and \(\nu_f\) does not increase
VAL \(\nu_f\) does not decrease
NUL \(\nu_f\) does not increase
THOL \(C(t)\) does not fall by more than 10%
ZHIR delegates phase, identity, and bifurcation checks to the mutation postcondition module
NAV at least one of \(\nu_f\), \(\theta\), or \(\Delta\mathrm{NFR}\) changes
REMESH advisory entry; returns success without a network-remesh check

The U3 phase-compatibility gate for UM and RA is a hard precondition in their operator pipeline, separate from the reactive postcondition table. Registry lookup uses the lower-case English function name ("coherence", "self_organization", and so on); a glyph string such as "IL" does not select the corresponding registry checker.

ZHIR temporal evidence and execution boundary

The Mutation contract owns trigger admission, history precedence, SDK abstention and identity/provenance semantics. The nodal product predicted_depi_dt = nu_f * DeltaNFR is an instantaneous prediction; it does not replace the observed signed two-sample secant required by the strict gate observed_depi_dt > xi. Active finite capacity and independent U4b context are also required.

Timestamped evidence must be finite, have a positive interval and end at the live EPI state. Invalid authoritative history cannot fall through to a legacy source. Untimestamped compatibility histories declare operator-step units, not elapsed physical time. Missing observations remain None; rate_gap is available only when observed and predicted rates share the physical-time basis. The event Mutation boundary adds timestamp/declared-duration agreement and execution provenance; these claims are separate from gate crossing.

The three-sample compute_d2epi_dt2 diagnostic and the branch-selection capacity threshold are also distinct from the two-sample trigger. Neither a predicted crossing, an acceleration alert nor a reactive postcondition establishes bifurcation, grammar readiness or future stability.

Registry coverage is therefore not a proof that every full operator contract has been verified. For a reproducible catalog-level measurement, use audit_operator_contracts. It builds controlled graphs, places each request in its intended context, checks the actually appended glyph so a grammar fallback cannot certify the request, and returns an OperatorContractAudit. Its REMESH case remains an advisory network-level result.

6.3 Monitor modes

Mode Behaviour
OFF Hook methods return default data without metric computation
OBSERVE Record reports and violations without raising
ENFORCE Raise StructuralIntegrityViolation on failure

6.4 Suggestions and SDK scope

Corrective suggestions are generated for recognized conservation-derived grammar labels, an increasing energy-candidate observation, or charge drift above the internal alert. A postcondition failure alone need not produce a suggestion.

The SDK exposes two different dictionary-returning conveniences:

  • Network.integrity_check(operator_name) calls after_operator directly for at most ten current nodes and returns operator, nodes_checked, passed, failed, pass_rate, and per-node reports. It does not execute an operator, capture a matching before snapshot, attach the monitor, or audit all 13 operators. Use an English function name to activate a registry check.
  • Network.audit_operators() runs the independent controlled audit_operator_contracts protocol and returns a dictionary with the 13 contextual results and its summary. It audits the operator implementation, rather than the current Network instance's trajectory.

Implementation: physics.integrity provides IntegrityReport, IntegritySummary, MonitorMode, StructuralIntegrityViolation, POSTCONDITIONS, and audit_operator_contracts.

Tests: tests/physics/test_structural_integrity.py and tests/sdk/test_simple_advanced.py.

Implementation and examples

Module Content
src/tnfr/physics/lyapunov.py Nominal U2-role multipliers and spectral diagnostics
src/tnfr/physics/structural_diffusion.py Fixed, time-varying and exact-common-metric pure-EPI flow certificates
src/tnfr/physics/hybrid_operator_stability.py Declared affine-reset gains and hybrid flow/reset budgets
src/tnfr/physics/reception_realization.py Read-only EN runtime-to-affine-flow boundary
src/tnfr/physics/resonance_realization.py Read-only identity-gated RA runtime-to-affine-flow boundary and post-RA metric audit
src/tnfr/physics/network_stage_stability.py All-target EN/RA certificates and the validated one-stage positive-duration post-flow bridge
src/tnfr/physics/pointwise_stage_stability.py Executor-bound pointwise affine realization and gain levels
src/tnfr/operators/event_timing.py Exact finite flow/jump schedules and binary64 clock readiness
src/tnfr/operators/event_runtime.py Atomic observed flow/glyph binding and finite represented EPI-map composition
src/tnfr/operators/event_remesh_runtime.py Atomic schedule/delayed-REMESH cycle with separate evidence channels
src/tnfr/operators/event_remesh_sequence.py Exact continuity across ordered supplied cycle results
src/tnfr/operators/event_remesh_causal_runtime.py / src/tnfr/operators/event_remesh_causal_runtime.pyi One graph-owned finite causal cycle sequence and exact public interface
src/tnfr/physics/event_refinement.py Offline and executor-linked event-local ZHIR evidence
src/tnfr/physics/event_remesh_refinement.py Finite strict three-mesh event/REMESH observations
src/tnfr/physics/event_remesh_reference.py / src/tnfr/physics/event_remesh_reference.pyi Effective-P2 finite reference-family certificate and exact public interface
src/tnfr/physics/reversible_eigenmode_reference.py / src/tnfr/physics/reversible_eigenmode_reference.pyi General exact-rational reversible single-eigenmode Euler theorem and public interface
src/tnfr/physics/runtime_eigenmode_reference.py / src/tnfr/physics/runtime_eigenmode_reference.pyi Finite executor binding with exact pressure/execution defects and full-matrix propagation
src/tnfr/physics/remesh_history_stability.py Exact uniform finite companion-history Lyapunov certificate
src/tnfr/physics/remesh_schedule_policy_stability.py / src/tnfr/physics/remesh_schedule_policy_stability.pyi Conditional exact uniform REMESH/schedule spatial-disagreement theorem
src/tnfr/physics/remesh_schedule_relative_defect_stability.py / src/tnfr/physics/remesh_schedule_relative_defect_stability.pyi Conditional exact q_eff=q*(1+eta) robust policy theorem
src/tnfr/physics/binary64_remesh_relative_defect.py / src/tnfr/physics/binary64_remesh_relative_defect.pyi Exact pairwise REMESH boundary; uniform alpha=1, eta=0 class; and sharp alpha=1/2 antisymmetric P2 class with eta=135/124
src/tnfr/physics/binary64_p2_reception_stability.py / src/tnfr/physics/binary64_p2_reception_stability.pyi Global q=0 P2 half-Reception kernel composed with the alpha=1 REMESH class
src/tnfr/physics/runtime_p2_reception_stage.py / src/tnfr/physics/runtime_p2_reception_stage.pyi Finite executor binding of one P2 two-phase EN EPI stage to the global q=0 kernel
src/tnfr/physics/runtime_p2_reception_remesh_sequence.py / src/tnfr/physics/runtime_p2_reception_remesh_sequence.pyi Finite causal binding of executed P2 EN/REMESH cycles and observed active-suffix extinction
src/tnfr/physics/runtime_p2_reception_remesh_policy.py / src/tnfr/physics/runtime_p2_reception_remesh_policy.pyi Transactional per-invocation P2 preflight, execution and finite certification
src/tnfr/physics/runtime_remesh_history_stability.py One-transition executed binary64 REMESH/companion bridge
src/tnfr/physics/remesh_schedule_stability.py Exact REMESH-head/schedule-head augmented-energy balance
src/tnfr/physics/runtime_remesh_schedule_stability.py Adjacent-cycle runtime/history energy telescope
src/tnfr/physics/runtime_remesh_schedule_block_margin.py / src/tnfr/physics/runtime_remesh_schedule_block_margin.pyi Exact normalized margin for a contiguous causally executed finite block
src/tnfr/physics/runtime_remesh_schedule_relative_defect.py / src/tnfr/physics/runtime_remesh_schedule_relative_defect.pyi Finite causal verification of signed defects and the robust policy envelope
src/tnfr/operators/_delayed_remesh_kernel.py Immutable delayed REMESH proposals and one-step evidence
src/tnfr/physics/phase_quotient.py Fixed-branch pairwise quotient, restricted canonical phase lift and counterexample
src/tnfr/physics/coherence_geometry.py Local, fixed-network and fixed-capacity coherence strata
src/tnfr/physics/phase_transition.py Order parameter, operational phase classification, effective exponent fit
src/tnfr/physics/life.py Strict supplied-series logistic diagnostics and selected \(A(t)>1\) event
src/tnfr/operators/lifecycle.py Instantaneous node-state and collapse predicates
src/tnfr/operators/hamiltonian.py Auxiliary matrix, unitary flow, spectrum, compatibility helpers
src/tnfr/physics/integrity.py Optional reactive monitor and contextual operator audit

The finite three-mesh contracts are falsified and sealed by test_event_remesh_refinement.py. The effective-P2 reference family and public example are checked by test_event_remesh_reference.py and test_event_remesh_reference_example.py. The general reversible exact-mode theorem, sealing and public P3 example are checked by test_reversible_eigenmode_reference.py and test_reversible_eigenmode_reference_example.py. The finite executor binding, proof seals and public executed P3 example are checked by test_runtime_eigenmode_reference.py and test_runtime_eigenmode_reference_example.py. The finite companion-history theorem is checked exactly by test_remesh_history_stability.py. The common-q exact policy theorem, public facade and example are checked by test_remesh_schedule_policy_stability.py and test_remesh_schedule_policy_stability_example.py. The robust signed-defect theorem and its finite causal adapter are checked by test_remesh_schedule_relative_defect_stability.py, test_runtime_remesh_schedule_relative_defect.py and test_runtime_remesh_schedule_relative_defect_example.py. The binary64 REMESH boundary, P2 numeric EPI-kernel composition and public examples are checked by test_binary64_remesh_relative_defect.py, test_half_alpha_antisymmetric_remesh_class.py, test_binary64_p2_reception_stability.py, test_binary64_remesh_relative_defect_example.py, test_half_alpha_antisymmetric_remesh_class_example.py and test_binary64_p2_reception_stability_example.py. The finite executed-stage binding and example are checked by test_runtime_p2_reception_stage.py and test_runtime_p2_reception_stage_example.py. The finite causal P2 EN/REMESH sequence, active-suffix boundary and public example are checked by test_runtime_p2_reception_remesh_sequence.py and test_runtime_p2_reception_remesh_sequence_example.py. The reusable policy, its rollback boundaries and two-call example are checked by test_runtime_p2_reception_remesh_policy.py and test_runtime_p2_reception_remesh_policy_example.py. The runtime residual and lifted-energy bridge is tested in test_runtime_remesh_history_stability.py. The pure schedule-head telescope and gain-based lower bound are tested in test_remesh_schedule_stability.py. The adjacent runtime schedule binding and finite history telescope are tested in test_runtime_remesh_schedule_stability.py. The finite same-invocation causal wrapper, receipt bindings, graph rollback and public example are tested in test_event_remesh_causal_runtime.py and test_event_remesh_causal_runtime_example.py. The exact finite-block identities, normalized diagnostics, public facade and 139/256 versus zero-margin examples are checked by test_runtime_remesh_schedule_block_margin.py and test_runtime_remesh_schedule_block_margin_example.py.

SDK Entry Points

from tnfr.physics.integrity import MonitorMode, enable_integrity_monitor
from tnfr.sdk import TNFR

net = TNFR.create(20).ring().evolve(5)
monitor = enable_integrity_monitor(net.G, mode=MonitorMode.OBSERVE)
# Subsequent operator-class calls append IntegrityReport objects to monitor.summary.

snapshot = net.integrity_check("coherence")  # dict; up to ten current nodes
catalog = net.audit_operators()               # dict; 13 controlled probes

Executable Demonstrations

Example Concept from this document
29_lyapunov_stability_demo.py Nominal operator-role multipliers, measured energy diagnostics, spectral read-outs, and life telemetry
161_core_research_trajectory.py Sampled pure-EPI path, modal limit, common energy budget and two-mesh comparison
162_hybrid_epi_stability.py Affine amplification absorbed by diffusion, consensus drift, and the infinite-gain local-offset witness
163_reception_runtime_bridge.py EN ideal-real, represented, runtime-snapshot and pressure-refresh boundary
164_resonance_runtime_bridge.py U3-filtered RA identity gate, four realization layers, post-RA flow certificate and switching abstention
166_event_remesh_reference_family.py Effective-P2 2/4/8-segment finite Euler/REMESH reference family and explicit false scope
167_reversible_eigenmode_reference.py Both exact nonuniform modes of nonregular P3, with exact-real Euler refinement bounds and explicit runtime abstention
168_runtime_reversible_eigenmode_reference.py Finite 2/4/8-segment executed nonregular-P3 binding with nonzero rho, eta, epsilon and explicit false runtime-convergence scope
169_event_remesh_causal_runtime.py One finite same-invocation event/REMESH cycle sequence with causal receipts, graph-owned atomicity, and the lag-one alpha=1 stability boundary
170_runtime_remesh_block_margin.py Exact kappa=139/256 finite-block lower margin and the causal alpha=1, kappa=0 boundary, with uniform-class and prefix claims withheld
171_remesh_schedule_policy_stability.py Conditional exact common-q policy theorem with prefix gain upper bound one, uniform block margin 1-q, strict pure-delay disagreement decay and the zero-margin q=1 boundary
172_runtime_remesh_relative_defect.py Exact q_eff=q*(1+eta) theorem bound to zero-defect and positive-binary64-defect finite causal blocks, with forward invariance and future stability withheld
173_binary64_remesh_relative_defect.py Exact pairwise alpha=1/2 obstruction and the forward-invariant REMESH-only alpha=1, eta=0 hard-clip class
174_binary64_p2_reception_remesh_stability.py Global q=0 P2 half-Reception numeric kernel composed with alpha=1, giving active-history extinction after tau_global+1 restricted cycles
175_runtime_p2_reception_stage.py One executed two-phase P2 EN EPI stage bound to the global q=0 kernel, with REMESH graph binding and repeated runtime withheld
176_runtime_p2_reception_remesh_sequence.py One completed same-invocation P2 EN/REMESH sequence with N >= tau_global+1 and finite observed active-history extinction
177_runtime_p2_reception_remesh_policy.py Two successive independently validated finite P2 policy invocations with future and auxiliary-state claims withheld
178_half_alpha_antisymmetric_remesh_class.py Sharp eta=135/124 for the REMESH-only alpha=1/2 antisymmetric P2 class, exact IEEE tail/core proof, strict and zero-margin q boundaries, and excluded generalizations

Cross-References